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# 6. Composite transformations

## Video transcript

did you get final approval from the director congratulations earlier we saw that translation and scaling don't commute let's see if we can get a better understanding of what's going on using some algebra suppose we translate by an amount of 5 + X + 3 + y pick a point x0 y0 in the image we're translating that point goes to a point x1 y1 given by x1 equals x0 + 5 y 1 equals y 0 + 3 now suppose we scale about the origin by a factor of 4 where does x1 y1 go let's call a point goes to x2 y2 scaling says x2 equals 4 times x1 and y2 equals 4 times y1 substitute our expressions for x1 and y1 x2 equals 4 times x0 + 5 which equals 4 times x0 + 4 times 5 which equals 4 times x0 plus 20 and y2 is equal to 4 times y 0 + 3 which equals 4 times y 0 + 12 this factor in front of x + y is 4 so the effective scale factor is still 4 however the effective translation amount is 20 + x + 12 + y for comparison let's do the operations in the opposite order scale first that is X 1 equals 4 times X 0 and y 1 equals 4 times y 0 then translate so algebraically x2 equals x1 plus 5 which equals 4 times X 0 plus 5 and Y 2 equals y1 plus 3 which equals 4 times y 0 plus 3 clearly the blue equations aren't the same as the red equations but in either case we can write the result of combining scaling and translation in the form x2 equals F times x0 + T and y2 equals s times y 0 plus T Y where T X stands for the effective or final translation amount in X and T Y is the effective translation amount in Y when two or more transformations are combined we call it a composite transformation in the next exercise you'll be asked to verify that this general form for composite transformation consisting of scales and translations always holds no matter how many scales and translations are combined and no matter what the order you